I'm trying to figure out how to uniquely distribute 2 red balls and 2 blue balls into 3 indistinguishable boxes.
My mind leapt to the general case, and I've found that there is a bijection between the number of ways we can uniquely distribute $k_1$ [color-one] balls, $k_2$ [color-two] balls, $k_3$ [color-three] balls, ... $k_f$ [color-$f$] balls into $n$ boxes and the number of ways to factor a product of primes into $n$ distinct factors.
More clearly, let $p_i$ give the $i$th prime. Then I'm looking for the number of ways to factor $$\prod_{i=1}^f p_i^{k_i}$$ into $n$ distinct factors. Call this number $\Phi$.
How can I approach this?
Edit: A bit more research revealed the following:
Let $\eta(k_1,k_2,...,k_f) = \prod_{i=1}^f p_i^{k_i}$. Also, let $Q(x)$ give the number of multiplicative partitions of a non-squarefree number $x$. Then we have
$$\Phi=\mu^2(\eta)\cdot B(f) + (1-\mu^2(\eta))\cdot Q(\eta)$$ where $\mu(n)$ denotes the Möbius function and $B(f)=\sum_{k=0}^n S(n,k) $ is the $f$th Bell number.
Now I just need to find a formula for $Q(x)$; could someone help me in this regard?